It is no doubt that the vacuous truth is related to material implication "$P\Rightarrow Q$". We say the material implication statement is true when $P$ is false. However, seems that this is not the definition for vacuously truth. Do we call it vacuously true only when $P$ can never be true? (Seems that all examples are in this way).

More clearly, suppose we have a statement "$P(x)\Rightarrow Q(x)$", and at some domains of $x$ (we may denote the domain by $T_x$), $P(x)$ is true; at some other domains of $x$ (denote the domain by $F_x$), $P(x)$ is false. Can we say the statement is vacuously true in the domain $F_x$? Is there any real example?

To make the problem more clear, we may check the statement "For all $x$, $P(x)\Rightarrow Q(x)$". Can we say this statement is vacuously true in the domain $F_x$ ($F_x$ is defined as above)?

  • $\begingroup$ Related. $\endgroup$ – user 170039 Sep 12 '18 at 17:46
  • $\begingroup$ @user170039 not really very related. I know why we call it true when $P$ is false. The question is that "is vacuously true more specific to that $P$ can never be true". $\endgroup$ – X liu Sep 12 '18 at 17:49
  • $\begingroup$ See Vacuous truth for both type of examples. $\endgroup$ – Mauro ALLEGRANZA Sep 12 '18 at 19:05
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    $\begingroup$ Example : $\forall x [(x < 0) \to (x > 0)]$, Is vacuously True in $\mathbb N$ and is False in $\mathbb Z$. $\endgroup$ – Mauro ALLEGRANZA Sep 12 '18 at 19:09
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    $\begingroup$ In FOL (first order logic) usually the domain in not specified into the formula. Every formula can be inetrpreted in different "contexts" (i.e. domain) : in some domain the formula is T, in other F, except for universally valid formulas, like e.g. $\forall x (x=x)$, that are T in every domain. $\endgroup$ – Mauro ALLEGRANZA Sep 13 '18 at 5:59

I would frame the concept as pertaining to first order sentences of the form $$\forall x(P(x)\to Q(x)).$$ Vacous truth is the particular situation when there are no $x$ in the domain satisfying $P(x),$ in which case standard semantics says the statement is true. So yes, whether a first order sentence is vacuously true or not depends on the interpretation, and hence the domain, much as regular truth does. (However this doesn’t usually matter: when we’re talking about math, we generally are making statements relative to some fixed interpretation.)

  • $\begingroup$ Sorry. I am not a student in math, can you make it more clear. What does 'first order sentences' mean? What is the fixed interpretation about vacuously true in math? We will not generally say it is vacuously true only when $P$ is false in math, am I right?Can we say $x\in T_x (P(x)\Rightarrow Q(x))$ vacuously true? $\endgroup$ – X liu Sep 12 '18 at 17:58
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    $\begingroup$ @xliu I will try to address what I think is your question. “If $2+2=5$, then I am the king of Sweden” is true, but not generally considered vacuously true. Vacuous truth involves “for all” statements. “Every murderer in the building is wearing a Christmas sweater” is vacuously true if there are no murderers in the building. It can also be true, but not vacuously true, if there are murderers in the building and they’re all wearing Christmas sweaters. $\endgroup$ – spaceisdarkgreen Sep 12 '18 at 18:36
  • $\begingroup$ That's exactly what I mean. The example in en.wikipedia.org/wiki/Vacuous_truth, "if Uluru is in France, then the Eiffel Tower is in Bolivia" is similar, but is treated vacuous true. There is no "for all" at all. So, I thought vacuous true is only for the case that "P" is never true. $\endgroup$ – X liu Sep 12 '18 at 19:38
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    $\begingroup$ @xliu yeah, I don’t use vacuously true that way (and note that it clashes with their initial statement of asserting all members of the empty set have a property), and in my experience it’s the most common usage. If other people also use it to refer to any implication with false antecedent, fine. It’s not usually defined formally for mathematical purposes, anyway. $\endgroup$ – spaceisdarkgreen Sep 12 '18 at 20:14

In my understanding if we want to use quantifier, the domain of discourse has to be specified first. This let us focus on those things we care the most. E.g. let $S$ be the set containing "all people not taller than 180cm", then $\bar S$ may contain a tree taller than 180, or a monkey who likes tomato, etc. Since the definition/description from Wikipedia uses quantifier:

["]In mathematics and logic, a vacuous truth is a statement that asserts that all members of the empty set have a certain property.["]

It depends on the domain of discourse, or the universal set, to say something is vacuously true. Using the just example following the definition

$$\textrm{all cell phones in the room are turned off.}$$

will be vacuously true whenever there are no cell phones in the room: Here a room with no cell phones is the domain of discourse, and in this case it's vacuously true. The adjective "vacuously" is omitted in the text.

But as the example you mentioned in the comment, when quantifier is not used, it can also be said vacuously true, and it's more formal (which is what Wiki says, not me)

More formally, a relatively well-defined usage refers to a conditional statement with a false antecedent.

In short, when

  • Quantifier is used, and the target $S$ is an empty (sub)set. (which leads to antecedent be true, because no counter-example can be found), more specifically
    $$[\forall x\in S, P(x)]\equiv T, \textrm{when }S=\emptyset$$ And this means the truth of $$[\forall x\in S, P(x)\implies Q]\equiv Q$$
    is depends on $Q$.
  • the antecedent (a single target) is false.

both are called vacuously true, because we don't care this case so much, we interested in "if $P$ happen, $Q$ will happen".

To make the problem more clear, we may check the statement "For all x, P(x)⇒Q(x)". Can we say this statement is vacuously true in the domain Fx (Fx is defined as above)?

Yes, only in the domain $F_x$.


Logicians typically define concepts such as these purely in terms of truth values rather than what's possible; this isn't a concept of modal logic. I think the examples you've seen (you didn't mention any specifically) probably just chose impossible $p$ so it's obvious to you that they're false.

  • $\begingroup$ I don't understand you answer clearly. So, if $P(x) $ can be right in some parts domain of $x$ and false in other part of domain. can we say "$P(x)\Rightarrow Q(x)$" is vacuously true in domain of $x$ where P(x) is false? $\endgroup$ – X liu Sep 12 '18 at 19:42
  • $\begingroup$ @Xliu You'd say for all $x$ if $P(x)$ then $Q(x)$, which spaceisdarkgreen has written formally. $\endgroup$ – J.G. Sep 12 '18 at 19:46
  • $\begingroup$ My idea is the statement with quantifiers (which is in fact a set of statements), so mathematically here the possibility in fact means the region of truth value . For first order logic, seems that the only quantifiers are $\forall$ and $\exists$. However, in general sense $P(x)$ and $Q(x)$ should have different truth values in different regions of $x$. In this sense, we may still discuss "possibility" in the terms of truth value. $\endgroup$ – X liu Sep 12 '18 at 20:51
  • $\begingroup$ @Xliu Or $P$ could be false everywhere; that's the vacuously true case. $\endgroup$ – J.G. Sep 13 '18 at 5:24

Do we call it vacuously true only when P can never be true?

No. In classical logic, for any logical propositions P and Q (it doesn't matter if they are true or false), we have:

$P\implies [\neg P \implies Q]$

This is a tautology. See truth table here.

  • $\begingroup$ Yes. I know it is true when $P$ is false. However, seems that not all mathematician call this true as vacuously true. See, for example @spaceisdarkgreen' s answer. And seems that it is important vacuously true is for all $\endgroup$ – X liu Sep 12 '18 at 23:11
  • $\begingroup$ It is also also true when P is true. See the truth table at the above link. This the most general form of vacuous proof. $\endgroup$ – Dan Christensen Sep 13 '18 at 1:18
  • $\begingroup$ It is true when P is false; it is true when P is true and Q is true; it is false when P is true and Q is false. However, this is not the point. The point is the definition of vacuous true for the statement with quantifiers for all. The case that "$P(x)$ is false for some specific $x$" is true for that $x$, but it is not vacuous true. See my discussions with @spaceisdarkgreen. $\endgroup$ – X liu Sep 13 '18 at 1:38
  • $\begingroup$ No, it's truth does not depend on the truth values of either P or Q. That's what we mean by a tautology. $P\implies [\neg P \implies Q]$ is always true. P and Q can be propositions in set theory, but need not be. They may also have universal quantifier within them, but not necessarily. $\endgroup$ – Dan Christensen Sep 13 '18 at 1:53
  • $\begingroup$ I agree with you that $P\Rightarrow[\neg P\Rightarrow Q]$ is always true. But, is this the definition of vacuously true? My question is what vacuously true should be? $\endgroup$ – X liu Sep 13 '18 at 2:13

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